solve algebraically using absolute value

Question

Solve algebraically: 

 

|x| + |x−2| > 5

Answer 1

Hi Bianca:

 

When dealing with absolute values you need to express the function y = |x| + |x-2| in different intervals.  The cut-off values for x are 0 and 2.

 

When x < 0, y = - x + (2-x) = 2(1-x)

 

When  0 < x < 2, y = x + (2-x) = 2

 

When x > 2, y = x + (x-2) = 2(x-1).

 

Now you can solve the inequalities:

 

When x < 0, 2(1-x) > 5  → -2x > 3 → x < -3/2 = -1.5

 

When 0 < x < 2, y = 2 > 5 is impossible.

 

When x > 2, 2(x-1) > 5 → 2x > 7 → x > 7/2 = 3.5

 

In summary, the solutions are:

x < -1.5  and x > 3.5

 

Hope this helps!

 

  Jean-Michel